None of the digits is over 7, indicating the numbers might very well be base-8. In base-10 the numbers seem to be rep-7's, so the table below lists the base-8 representation of the first fifteen numbers that consist solely of 7's in base 10.

From the table we see the next value, represented by the question mark is 450545561.

Larry's previously posted problem's involvement with other another base (for pi) was a helpful hint in directing thinking toward, this time, base 8. 

7    7

77    115

777    1411

7777    17141

77777    227721

777777    2757061

7777777    35526761

77777777    450545561

777777777    5626771161

7777777777    71745674161

77777777777    1103372532161

777777777777    13242712606161

7777777777777    161134753476161

77777777777777    2153641464156161

777777777777777    26066120012116161

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 Text1.Text = ""

 crlf = Chr(13) + Chr(10)

 

 For i = 1 To 15

   n = n * 10 + 7

   Text1.Text = Text1.Text & n & "    " & base$(n, 8) & crlf

 Next i

 

 

 

 

 

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub

Function base$(n, b)

  v$ = ""

  n2 = n

  Do

    q = Int(n2 / b)

    d = n2 - q * b

    n2 = q

    v$ = Mid("0123456789abcdefghijklmnopqrstuvwxyz", d + 1, 1) + v$

  Loop Until n2 = 0

  base$ = v$

End Function

Function fromBase(n$, b)

  v = 0

  For i = 1 To Len(n$)

    c$ = LCase$(Mid(n$, i, 1))

    If c$ > " " Then

      v = v * b + InStr("0123456789abcdefghijklmnopqrstuvwxyz", c$) - 1

    End If

  Next

  fromBase = v

End Function